Properties

Label 5586.2.a.t.1.1
Level $5586$
Weight $2$
Character 5586.1
Self dual yes
Analytic conductor $44.604$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5586 = 2 \cdot 3 \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5586.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.6044345691\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 798)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5586.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -1.00000 q^{12} -2.00000 q^{13} +2.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} -1.00000 q^{19} -2.00000 q^{20} +4.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} -2.00000 q^{29} +2.00000 q^{30} +1.00000 q^{32} +2.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} -1.00000 q^{38} +2.00000 q^{39} -2.00000 q^{40} +6.00000 q^{41} +4.00000 q^{43} -2.00000 q^{45} +4.00000 q^{46} -1.00000 q^{48} -1.00000 q^{50} -2.00000 q^{51} -2.00000 q^{52} -10.0000 q^{53} -1.00000 q^{54} +1.00000 q^{57} -2.00000 q^{58} -12.0000 q^{59} +2.00000 q^{60} +10.0000 q^{61} +1.00000 q^{64} +4.00000 q^{65} -8.00000 q^{67} +2.00000 q^{68} -4.00000 q^{69} +1.00000 q^{72} +6.00000 q^{73} -2.00000 q^{74} +1.00000 q^{75} -1.00000 q^{76} +2.00000 q^{78} -4.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -4.00000 q^{83} -4.00000 q^{85} +4.00000 q^{86} +2.00000 q^{87} -10.0000 q^{89} -2.00000 q^{90} +4.00000 q^{92} +2.00000 q^{95} -1.00000 q^{96} +2.00000 q^{97} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −2.00000 −0.632456
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 1.00000 0.235702
\(19\) −1.00000 −0.229416
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) −1.00000 −0.204124
\(25\) −1.00000 −0.200000
\(26\) −2.00000 −0.392232
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 2.00000 0.365148
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −1.00000 −0.162221
\(39\) 2.00000 0.320256
\(40\) −2.00000 −0.316228
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) −2.00000 −0.298142
\(46\) 4.00000 0.589768
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) −1.00000 −0.141421
\(51\) −2.00000 −0.280056
\(52\) −2.00000 −0.277350
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) −2.00000 −0.262613
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 2.00000 0.258199
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 2.00000 0.242536
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.00000 0.117851
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) −2.00000 −0.232495
\(75\) 1.00000 0.115470
\(76\) −1.00000 −0.114708
\(77\) 0 0
\(78\) 2.00000 0.226455
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) −2.00000 −0.223607
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) 4.00000 0.431331
\(87\) 2.00000 0.214423
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) −2.00000 −0.210819
\(91\) 0 0
\(92\) 4.00000 0.417029
\(93\) 0 0
\(94\) 0 0
\(95\) 2.00000 0.205196
\(96\) −1.00000 −0.102062
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) −2.00000 −0.198030
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) −10.0000 −0.971286
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 1.00000 0.0936586
\(115\) −8.00000 −0.746004
\(116\) −2.00000 −0.185695
\(117\) −2.00000 −0.184900
\(118\) −12.0000 −1.10469
\(119\) 0 0
\(120\) 2.00000 0.182574
\(121\) −11.0000 −1.00000
\(122\) 10.0000 0.905357
\(123\) −6.00000 −0.541002
\(124\) 0 0
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.00000 −0.352180
\(130\) 4.00000 0.350823
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −8.00000 −0.691095
\(135\) 2.00000 0.172133
\(136\) 2.00000 0.171499
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) −4.00000 −0.340503
\(139\) 20.0000 1.69638 0.848189 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 4.00000 0.332182
\(146\) 6.00000 0.496564
\(147\) 0 0
\(148\) −2.00000 −0.164399
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 1.00000 0.0816497
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) −14.0000 −1.11732 −0.558661 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) −4.00000 −0.318223
\(159\) 10.0000 0.793052
\(160\) −2.00000 −0.158114
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −20.0000 −1.56652 −0.783260 0.621694i \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −4.00000 −0.306786
\(171\) −1.00000 −0.0764719
\(172\) 4.00000 0.304997
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 2.00000 0.151620
\(175\) 0 0
\(176\) 0 0
\(177\) 12.0000 0.901975
\(178\) −10.0000 −0.749532
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) −2.00000 −0.149071
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) −10.0000 −0.739221
\(184\) 4.00000 0.294884
\(185\) 4.00000 0.294086
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 2.00000 0.145095
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 2.00000 0.143592
\(195\) −4.00000 −0.286446
\(196\) 0 0
\(197\) −10.0000 −0.712470 −0.356235 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 8.00000 0.564276
\(202\) −2.00000 −0.140720
\(203\) 0 0
\(204\) −2.00000 −0.140028
\(205\) −12.0000 −0.838116
\(206\) −8.00000 −0.557386
\(207\) 4.00000 0.278019
\(208\) −2.00000 −0.138675
\(209\) 0 0
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) −10.0000 −0.686803
\(213\) 0 0
\(214\) −4.00000 −0.273434
\(215\) −8.00000 −0.545595
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −10.0000 −0.677285
\(219\) −6.00000 −0.405442
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) 2.00000 0.134231
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 2.00000 0.133038
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 1.00000 0.0662266
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) −8.00000 −0.527504
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) 4.00000 0.259828
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 2.00000 0.129099
\(241\) −6.00000 −0.386494 −0.193247 0.981150i \(-0.561902\pi\)
−0.193247 + 0.981150i \(0.561902\pi\)
\(242\) −11.0000 −0.707107
\(243\) −1.00000 −0.0641500
\(244\) 10.0000 0.640184
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) 2.00000 0.127257
\(248\) 0 0
\(249\) 4.00000 0.253490
\(250\) 12.0000 0.758947
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 4.00000 0.250982
\(255\) 4.00000 0.250490
\(256\) 1.00000 0.0625000
\(257\) −10.0000 −0.623783 −0.311891 0.950118i \(-0.600963\pi\)
−0.311891 + 0.950118i \(0.600963\pi\)
\(258\) −4.00000 −0.249029
\(259\) 0 0
\(260\) 4.00000 0.248069
\(261\) −2.00000 −0.123797
\(262\) −12.0000 −0.741362
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 0 0
\(265\) 20.0000 1.22859
\(266\) 0 0
\(267\) 10.0000 0.611990
\(268\) −8.00000 −0.488678
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 2.00000 0.121716
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) 2.00000 0.120824
\(275\) 0 0
\(276\) −4.00000 −0.240772
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 20.0000 1.19952
\(279\) 0 0
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) −12.0000 −0.713326 −0.356663 0.934233i \(-0.616086\pi\)
−0.356663 + 0.934233i \(0.616086\pi\)
\(284\) 0 0
\(285\) −2.00000 −0.118470
\(286\) 0 0
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) 4.00000 0.234888
\(291\) −2.00000 −0.117242
\(292\) 6.00000 0.351123
\(293\) 2.00000 0.116841 0.0584206 0.998292i \(-0.481394\pi\)
0.0584206 + 0.998292i \(0.481394\pi\)
\(294\) 0 0
\(295\) 24.0000 1.39733
\(296\) −2.00000 −0.116248
\(297\) 0 0
\(298\) 6.00000 0.347571
\(299\) −8.00000 −0.462652
\(300\) 1.00000 0.0577350
\(301\) 0 0
\(302\) −12.0000 −0.690522
\(303\) 2.00000 0.114897
\(304\) −1.00000 −0.0573539
\(305\) −20.0000 −1.14520
\(306\) 2.00000 0.114332
\(307\) 4.00000 0.228292 0.114146 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 2.00000 0.113228
\(313\) −26.0000 −1.46961 −0.734803 0.678280i \(-0.762726\pi\)
−0.734803 + 0.678280i \(0.762726\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) 30.0000 1.68497 0.842484 0.538721i \(-0.181092\pi\)
0.842484 + 0.538721i \(0.181092\pi\)
\(318\) 10.0000 0.560772
\(319\) 0 0
\(320\) −2.00000 −0.111803
\(321\) 4.00000 0.223258
\(322\) 0 0
\(323\) −2.00000 −0.111283
\(324\) 1.00000 0.0555556
\(325\) 2.00000 0.110940
\(326\) −20.0000 −1.10770
\(327\) 10.0000 0.553001
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) −8.00000 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) −4.00000 −0.219529
\(333\) −2.00000 −0.109599
\(334\) 0 0
\(335\) 16.0000 0.874173
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) −9.00000 −0.489535
\(339\) −2.00000 −0.108625
\(340\) −4.00000 −0.216930
\(341\) 0 0
\(342\) −1.00000 −0.0540738
\(343\) 0 0
\(344\) 4.00000 0.215666
\(345\) 8.00000 0.430706
\(346\) −6.00000 −0.322562
\(347\) 24.0000 1.28839 0.644194 0.764862i \(-0.277193\pi\)
0.644194 + 0.764862i \(0.277193\pi\)
\(348\) 2.00000 0.107211
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 12.0000 0.637793
\(355\) 0 0
\(356\) −10.0000 −0.529999
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 28.0000 1.47778 0.738892 0.673824i \(-0.235349\pi\)
0.738892 + 0.673824i \(0.235349\pi\)
\(360\) −2.00000 −0.105409
\(361\) 1.00000 0.0526316
\(362\) −2.00000 −0.105118
\(363\) 11.0000 0.577350
\(364\) 0 0
\(365\) −12.0000 −0.628109
\(366\) −10.0000 −0.522708
\(367\) −32.0000 −1.67039 −0.835193 0.549957i \(-0.814644\pi\)
−0.835193 + 0.549957i \(0.814644\pi\)
\(368\) 4.00000 0.208514
\(369\) 6.00000 0.312348
\(370\) 4.00000 0.207950
\(371\) 0 0
\(372\) 0 0
\(373\) 6.00000 0.310668 0.155334 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) 0 0
\(375\) −12.0000 −0.619677
\(376\) 0 0
\(377\) 4.00000 0.206010
\(378\) 0 0
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 2.00000 0.102598
\(381\) −4.00000 −0.204926
\(382\) 12.0000 0.613973
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) 4.00000 0.203331
\(388\) 2.00000 0.101535
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) −4.00000 −0.202548
\(391\) 8.00000 0.404577
\(392\) 0 0
\(393\) 12.0000 0.605320
\(394\) −10.0000 −0.503793
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) −30.0000 −1.50566 −0.752828 0.658217i \(-0.771311\pi\)
−0.752828 + 0.658217i \(0.771311\pi\)
\(398\) −16.0000 −0.802008
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 8.00000 0.399004
\(403\) 0 0
\(404\) −2.00000 −0.0995037
\(405\) −2.00000 −0.0993808
\(406\) 0 0
\(407\) 0 0
\(408\) −2.00000 −0.0990148
\(409\) −22.0000 −1.08783 −0.543915 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) −12.0000 −0.592638
\(411\) −2.00000 −0.0986527
\(412\) −8.00000 −0.394132
\(413\) 0 0
\(414\) 4.00000 0.196589
\(415\) 8.00000 0.392705
\(416\) −2.00000 −0.0980581
\(417\) −20.0000 −0.979404
\(418\) 0 0
\(419\) −28.0000 −1.36789 −0.683945 0.729534i \(-0.739737\pi\)
−0.683945 + 0.729534i \(0.739737\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) −8.00000 −0.389434
\(423\) 0 0
\(424\) −10.0000 −0.485643
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) 0 0
\(428\) −4.00000 −0.193347
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) −4.00000 −0.191785
\(436\) −10.0000 −0.478913
\(437\) −4.00000 −0.191346
\(438\) −6.00000 −0.286691
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −4.00000 −0.190261
\(443\) 32.0000 1.52037 0.760183 0.649709i \(-0.225109\pi\)
0.760183 + 0.649709i \(0.225109\pi\)
\(444\) 2.00000 0.0949158
\(445\) 20.0000 0.948091
\(446\) 8.00000 0.378811
\(447\) −6.00000 −0.283790
\(448\) 0 0
\(449\) −14.0000 −0.660701 −0.330350 0.943858i \(-0.607167\pi\)
−0.330350 + 0.943858i \(0.607167\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 0 0
\(452\) 2.00000 0.0940721
\(453\) 12.0000 0.563809
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) 26.0000 1.21623 0.608114 0.793849i \(-0.291926\pi\)
0.608114 + 0.793849i \(0.291926\pi\)
\(458\) −14.0000 −0.654177
\(459\) −2.00000 −0.0933520
\(460\) −8.00000 −0.373002
\(461\) −10.0000 −0.465746 −0.232873 0.972507i \(-0.574813\pi\)
−0.232873 + 0.972507i \(0.574813\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) 18.0000 0.833834
\(467\) −4.00000 −0.185098 −0.0925490 0.995708i \(-0.529501\pi\)
−0.0925490 + 0.995708i \(0.529501\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 0 0
\(470\) 0 0
\(471\) 14.0000 0.645086
\(472\) −12.0000 −0.552345
\(473\) 0 0
\(474\) 4.00000 0.183726
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) −10.0000 −0.457869
\(478\) 12.0000 0.548867
\(479\) −8.00000 −0.365529 −0.182765 0.983157i \(-0.558505\pi\)
−0.182765 + 0.983157i \(0.558505\pi\)
\(480\) 2.00000 0.0912871
\(481\) 4.00000 0.182384
\(482\) −6.00000 −0.273293
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) −4.00000 −0.181631
\(486\) −1.00000 −0.0453609
\(487\) 12.0000 0.543772 0.271886 0.962329i \(-0.412353\pi\)
0.271886 + 0.962329i \(0.412353\pi\)
\(488\) 10.0000 0.452679
\(489\) 20.0000 0.904431
\(490\) 0 0
\(491\) −40.0000 −1.80517 −0.902587 0.430507i \(-0.858335\pi\)
−0.902587 + 0.430507i \(0.858335\pi\)
\(492\) −6.00000 −0.270501
\(493\) −4.00000 −0.180151
\(494\) 2.00000 0.0899843
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 4.00000 0.179244
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 12.0000 0.536656
\(501\) 0 0
\(502\) 12.0000 0.535586
\(503\) 32.0000 1.42681 0.713405 0.700752i \(-0.247152\pi\)
0.713405 + 0.700752i \(0.247152\pi\)
\(504\) 0 0
\(505\) 4.00000 0.177998
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) 4.00000 0.177471
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 4.00000 0.177123
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 1.00000 0.0441511
\(514\) −10.0000 −0.441081
\(515\) 16.0000 0.705044
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 4.00000 0.175412
\(521\) 22.0000 0.963837 0.481919 0.876216i \(-0.339940\pi\)
0.481919 + 0.876216i \(0.339940\pi\)
\(522\) −2.00000 −0.0875376
\(523\) −36.0000 −1.57417 −0.787085 0.616844i \(-0.788411\pi\)
−0.787085 + 0.616844i \(0.788411\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) 12.0000 0.523225
\(527\) 0 0
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 20.0000 0.868744
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) −12.0000 −0.519778
\(534\) 10.0000 0.432742
\(535\) 8.00000 0.345870
\(536\) −8.00000 −0.345547
\(537\) 12.0000 0.517838
\(538\) −6.00000 −0.258678
\(539\) 0 0
\(540\) 2.00000 0.0860663
\(541\) 30.0000 1.28980 0.644900 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) −16.0000 −0.687259
\(543\) 2.00000 0.0858282
\(544\) 2.00000 0.0857493
\(545\) 20.0000 0.856706
\(546\) 0 0
\(547\) −8.00000 −0.342055 −0.171028 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) 2.00000 0.0854358
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) 2.00000 0.0852029
\(552\) −4.00000 −0.170251
\(553\) 0 0
\(554\) −10.0000 −0.424859
\(555\) −4.00000 −0.169791
\(556\) 20.0000 0.848189
\(557\) −2.00000 −0.0847427 −0.0423714 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) −6.00000 −0.253095
\(563\) 44.0000 1.85438 0.927189 0.374593i \(-0.122217\pi\)
0.927189 + 0.374593i \(0.122217\pi\)
\(564\) 0 0
\(565\) −4.00000 −0.168281
\(566\) −12.0000 −0.504398
\(567\) 0 0
\(568\) 0 0
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) −2.00000 −0.0837708
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) 0 0
\(573\) −12.0000 −0.501307
\(574\) 0 0
\(575\) −4.00000 −0.166812
\(576\) 1.00000 0.0416667
\(577\) −34.0000 −1.41544 −0.707719 0.706494i \(-0.750276\pi\)
−0.707719 + 0.706494i \(0.750276\pi\)
\(578\) −13.0000 −0.540729
\(579\) 14.0000 0.581820
\(580\) 4.00000 0.166091
\(581\) 0 0
\(582\) −2.00000 −0.0829027
\(583\) 0 0
\(584\) 6.00000 0.248282
\(585\) 4.00000 0.165380
\(586\) 2.00000 0.0826192
\(587\) −44.0000 −1.81607 −0.908037 0.418890i \(-0.862419\pi\)
−0.908037 + 0.418890i \(0.862419\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 24.0000 0.988064
\(591\) 10.0000 0.411345
\(592\) −2.00000 −0.0821995
\(593\) 42.0000 1.72473 0.862367 0.506284i \(-0.168981\pi\)
0.862367 + 0.506284i \(0.168981\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) 16.0000 0.654836
\(598\) −8.00000 −0.327144
\(599\) 16.0000 0.653742 0.326871 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) 1.00000 0.0408248
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 0 0
\(603\) −8.00000 −0.325785
\(604\) −12.0000 −0.488273
\(605\) 22.0000 0.894427
\(606\) 2.00000 0.0812444
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) −20.0000 −0.809776
\(611\) 0 0
\(612\) 2.00000 0.0808452
\(613\) 6.00000 0.242338 0.121169 0.992632i \(-0.461336\pi\)
0.121169 + 0.992632i \(0.461336\pi\)
\(614\) 4.00000 0.161427
\(615\) 12.0000 0.483887
\(616\) 0 0
\(617\) −30.0000 −1.20775 −0.603877 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) 8.00000 0.321807
\(619\) 12.0000 0.482321 0.241160 0.970485i \(-0.422472\pi\)
0.241160 + 0.970485i \(0.422472\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) 0 0
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) −19.0000 −0.760000
\(626\) −26.0000 −1.03917
\(627\) 0 0
\(628\) −14.0000 −0.558661
\(629\) −4.00000 −0.159490
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) −4.00000 −0.159111
\(633\) 8.00000 0.317971
\(634\) 30.0000 1.19145
\(635\) −8.00000 −0.317470
\(636\) 10.0000 0.396526
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) −2.00000 −0.0790569
\(641\) −46.0000 −1.81689 −0.908445 0.418004i \(-0.862730\pi\)
−0.908445 + 0.418004i \(0.862730\pi\)
\(642\) 4.00000 0.157867
\(643\) 20.0000 0.788723 0.394362 0.918955i \(-0.370966\pi\)
0.394362 + 0.918955i \(0.370966\pi\)
\(644\) 0 0
\(645\) 8.00000 0.315000
\(646\) −2.00000 −0.0786889
\(647\) −16.0000 −0.629025 −0.314512 0.949253i \(-0.601841\pi\)
−0.314512 + 0.949253i \(0.601841\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) −20.0000 −0.783260
\(653\) 6.00000 0.234798 0.117399 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) 10.0000 0.391031
\(655\) 24.0000 0.937758
\(656\) 6.00000 0.234261
\(657\) 6.00000 0.234082
\(658\) 0 0
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 0 0
\(661\) −2.00000 −0.0777910 −0.0388955 0.999243i \(-0.512384\pi\)
−0.0388955 + 0.999243i \(0.512384\pi\)
\(662\) −8.00000 −0.310929
\(663\) 4.00000 0.155347
\(664\) −4.00000 −0.155230
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) −8.00000 −0.309761
\(668\) 0 0
\(669\) −8.00000 −0.309298
\(670\) 16.0000 0.618134
\(671\) 0 0
\(672\) 0 0
\(673\) 10.0000 0.385472 0.192736 0.981251i \(-0.438264\pi\)
0.192736 + 0.981251i \(0.438264\pi\)
\(674\) −22.0000 −0.847408
\(675\) 1.00000 0.0384900
\(676\) −9.00000 −0.346154
\(677\) 10.0000 0.384331 0.192166 0.981363i \(-0.438449\pi\)
0.192166 + 0.981363i \(0.438449\pi\)
\(678\) −2.00000 −0.0768095
\(679\) 0 0
\(680\) −4.00000 −0.153393
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) 4.00000 0.153056 0.0765279 0.997067i \(-0.475617\pi\)
0.0765279 + 0.997067i \(0.475617\pi\)
\(684\) −1.00000 −0.0382360
\(685\) −4.00000 −0.152832
\(686\) 0 0
\(687\) 14.0000 0.534133
\(688\) 4.00000 0.152499
\(689\) 20.0000 0.761939
\(690\) 8.00000 0.304555
\(691\) 12.0000 0.456502 0.228251 0.973602i \(-0.426699\pi\)
0.228251 + 0.973602i \(0.426699\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) 24.0000 0.911028
\(695\) −40.0000 −1.51729
\(696\) 2.00000 0.0758098
\(697\) 12.0000 0.454532
\(698\) 2.00000 0.0757011
\(699\) −18.0000 −0.680823
\(700\) 0 0
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 2.00000 0.0754851
\(703\) 2.00000 0.0754314
\(704\) 0 0
\(705\) 0 0
\(706\) 18.0000 0.677439
\(707\) 0 0
\(708\) 12.0000 0.450988
\(709\) 22.0000 0.826227 0.413114 0.910679i \(-0.364441\pi\)
0.413114 + 0.910679i \(0.364441\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) −10.0000 −0.374766
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) −12.0000 −0.448148
\(718\) 28.0000 1.04495
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) −2.00000 −0.0745356
\(721\) 0 0
\(722\) 1.00000 0.0372161
\(723\) 6.00000 0.223142
\(724\) −2.00000 −0.0743294
\(725\) 2.00000 0.0742781
\(726\) 11.0000 0.408248
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −12.0000 −0.444140
\(731\) 8.00000 0.295891
\(732\) −10.0000 −0.369611
\(733\) 50.0000 1.84679 0.923396 0.383849i \(-0.125402\pi\)
0.923396 + 0.383849i \(0.125402\pi\)
\(734\) −32.0000 −1.18114
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) 0 0
\(738\) 6.00000 0.220863
\(739\) 4.00000 0.147142 0.0735712 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 4.00000 0.147043
\(741\) −2.00000 −0.0734718
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) −12.0000 −0.439646
\(746\) 6.00000 0.219676
\(747\) −4.00000 −0.146352
\(748\) 0 0
\(749\) 0 0
\(750\) −12.0000 −0.438178
\(751\) −12.0000 −0.437886 −0.218943 0.975738i \(-0.570261\pi\)
−0.218943 + 0.975738i \(0.570261\pi\)
\(752\) 0 0
\(753\) −12.0000 −0.437304
\(754\) 4.00000 0.145671
\(755\) 24.0000 0.873449
\(756\) 0 0
\(757\) 22.0000 0.799604 0.399802 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) 8.00000 0.290573
\(759\) 0 0
\(760\) 2.00000 0.0725476
\(761\) 50.0000 1.81250 0.906249 0.422744i \(-0.138933\pi\)
0.906249 + 0.422744i \(0.138933\pi\)
\(762\) −4.00000 −0.144905
\(763\) 0 0
\(764\) 12.0000 0.434145
\(765\) −4.00000 −0.144620
\(766\) 0 0
\(767\) 24.0000 0.866590
\(768\) −1.00000 −0.0360844
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) 10.0000 0.360141
\(772\) −14.0000 −0.503871
\(773\) 18.0000 0.647415 0.323708 0.946157i \(-0.395071\pi\)
0.323708 + 0.946157i \(0.395071\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) −18.0000 −0.645331
\(779\) −6.00000 −0.214972
\(780\) −4.00000 −0.143223
\(781\) 0 0
\(782\) 8.00000 0.286079
\(783\) 2.00000 0.0714742
\(784\) 0 0
\(785\) 28.0000 0.999363
\(786\) 12.0000 0.428026
\(787\) 52.0000 1.85360 0.926800 0.375555i \(-0.122548\pi\)
0.926800 + 0.375555i \(0.122548\pi\)
\(788\) −10.0000 −0.356235
\(789\) −12.0000 −0.427211
\(790\) 8.00000 0.284627
\(791\) 0 0
\(792\) 0 0
\(793\) −20.0000 −0.710221
\(794\) −30.0000 −1.06466
\(795\) −20.0000 −0.709327
\(796\) −16.0000 −0.567105
\(797\) −6.00000 −0.212531 −0.106265 0.994338i \(-0.533889\pi\)
−0.106265 + 0.994338i \(0.533889\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) −10.0000 −0.353333
\(802\) 18.0000 0.635602
\(803\) 0 0
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) 0 0
\(807\) 6.00000 0.211210
\(808\) −2.00000 −0.0703598
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) −2.00000 −0.0702728
\(811\) 44.0000 1.54505 0.772524 0.634985i \(-0.218994\pi\)
0.772524 + 0.634985i \(0.218994\pi\)
\(812\) 0 0
\(813\) 16.0000 0.561144
\(814\) 0 0
\(815\) 40.0000 1.40114
\(816\) −2.00000 −0.0700140
\(817\) −4.00000 −0.139942
\(818\) −22.0000 −0.769212
\(819\) 0 0
\(820\) −12.0000 −0.419058
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) −2.00000 −0.0697580
\(823\) −8.00000 −0.278862 −0.139431 0.990232i \(-0.544527\pi\)
−0.139431 + 0.990232i \(0.544527\pi\)
\(824\) −8.00000 −0.278693
\(825\) 0 0
\(826\) 0 0
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 4.00000 0.139010
\(829\) 14.0000 0.486240 0.243120 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) 8.00000 0.277684
\(831\) 10.0000 0.346896
\(832\) −2.00000 −0.0693375
\(833\) 0 0
\(834\) −20.0000 −0.692543
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) −28.0000 −0.967244
\(839\) −16.0000 −0.552381 −0.276191 0.961103i \(-0.589072\pi\)
−0.276191 + 0.961103i \(0.589072\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −10.0000 −0.344623
\(843\) 6.00000 0.206651
\(844\) −8.00000 −0.275371
\(845\) 18.0000 0.619219
\(846\) 0 0
\(847\) 0 0
\(848\) −10.0000 −0.343401
\(849\) 12.0000 0.411839
\(850\) −2.00000 −0.0685994
\(851\) −8.00000 −0.274236
\(852\) 0 0
\(853\) −30.0000 −1.02718 −0.513590 0.858036i \(-0.671685\pi\)
−0.513590 + 0.858036i \(0.671685\pi\)
\(854\) 0 0
\(855\) 2.00000 0.0683986
\(856\) −4.00000 −0.136717
\(857\) 6.00000 0.204956 0.102478 0.994735i \(-0.467323\pi\)
0.102478 + 0.994735i \(0.467323\pi\)
\(858\) 0 0
\(859\) 28.0000 0.955348 0.477674 0.878537i \(-0.341480\pi\)
0.477674 + 0.878537i \(0.341480\pi\)
\(860\) −8.00000 −0.272798
\(861\) 0 0
\(862\) 24.0000 0.817443
\(863\) −48.0000 −1.63394 −0.816970 0.576681i \(-0.804348\pi\)
−0.816970 + 0.576681i \(0.804348\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 12.0000 0.408012
\(866\) 2.00000 0.0679628
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) 0 0
\(870\) −4.00000 −0.135613
\(871\) 16.0000 0.542139
\(872\) −10.0000 −0.338643
\(873\) 2.00000 0.0676897
\(874\) −4.00000 −0.135302
\(875\) 0 0
\(876\) −6.00000 −0.202721
\(877\) −58.0000 −1.95852 −0.979260 0.202606i \(-0.935059\pi\)
−0.979260 + 0.202606i \(0.935059\pi\)
\(878\) 8.00000 0.269987
\(879\) −2.00000 −0.0674583
\(880\) 0 0
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 0 0
\(883\) −4.00000 −0.134611 −0.0673054 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) −4.00000 −0.134535
\(885\) −24.0000 −0.806751
\(886\) 32.0000 1.07506
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) 2.00000 0.0671156
\(889\) 0 0
\(890\) 20.0000 0.670402
\(891\) 0 0
\(892\) 8.00000 0.267860
\(893\) 0 0
\(894\) −6.00000 −0.200670
\(895\) 24.0000 0.802232
\(896\) 0 0
\(897\) 8.00000 0.267112
\(898\) −14.0000 −0.467186
\(899\) 0 0
\(900\) −1.00000 −0.0333333
\(901\) −20.0000 −0.666297
\(902\) 0 0
\(903\) 0 0
\(904\) 2.00000 0.0665190
\(905\) 4.00000 0.132964
\(906\) 12.0000 0.398673
\(907\) −8.00000 −0.265636 −0.132818 0.991140i \(-0.542403\pi\)
−0.132818 + 0.991140i \(0.542403\pi\)
\(908\) −12.0000 −0.398234
\(909\) −2.00000 −0.0663358
\(910\) 0 0
\(911\) 24.0000 0.795155 0.397578 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) 1.00000 0.0331133
\(913\) 0 0
\(914\) 26.0000 0.860004
\(915\) 20.0000 0.661180
\(916\) −14.0000 −0.462573
\(917\) 0 0
\(918\) −2.00000 −0.0660098
\(919\) −40.0000 −1.31948 −0.659739 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) −8.00000 −0.263752
\(921\) −4.00000 −0.131804
\(922\) −10.0000 −0.329332
\(923\) 0 0
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) 16.0000 0.525793
\(927\) −8.00000 −0.262754
\(928\) −2.00000 −0.0656532
\(929\) 2.00000 0.0656179 0.0328089 0.999462i \(-0.489555\pi\)
0.0328089 + 0.999462i \(0.489555\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 18.0000 0.589610
\(933\) 0 0
\(934\) −4.00000 −0.130884
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) −34.0000 −1.11073 −0.555366 0.831606i \(-0.687422\pi\)
−0.555366 + 0.831606i \(0.687422\pi\)
\(938\) 0 0
\(939\) 26.0000 0.848478
\(940\) 0 0
\(941\) 10.0000 0.325991 0.162995 0.986627i \(-0.447884\pi\)
0.162995 + 0.986627i \(0.447884\pi\)
\(942\) 14.0000 0.456145
\(943\) 24.0000 0.781548
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 4.00000 0.129914
\(949\) −12.0000 −0.389536
\(950\) 1.00000 0.0324443
\(951\) −30.0000 −0.972817
\(952\) 0 0
\(953\) −54.0000 −1.74923 −0.874616 0.484817i \(-0.838886\pi\)
−0.874616 + 0.484817i \(0.838886\pi\)
\(954\) −10.0000 −0.323762
\(955\) −24.0000 −0.776622
\(956\) 12.0000 0.388108
\(957\) 0 0
\(958\) −8.00000 −0.258468
\(959\) 0 0
\(960\) 2.00000 0.0645497
\(961\) −31.0000 −1.00000
\(962\) 4.00000 0.128965
\(963\) −4.00000 −0.128898
\(964\) −6.00000 −0.193247
\(965\) 28.0000 0.901352
\(966\) 0 0
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) −11.0000 −0.353553
\(969\) 2.00000 0.0642493
\(970\) −4.00000 −0.128432
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 12.0000 0.384505
\(975\) −2.00000 −0.0640513
\(976\) 10.0000 0.320092
\(977\) 50.0000 1.59964 0.799821 0.600239i \(-0.204928\pi\)
0.799821 + 0.600239i \(0.204928\pi\)
\(978\) 20.0000 0.639529
\(979\) 0 0
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) −40.0000 −1.27645
\(983\) −16.0000 −0.510321 −0.255160 0.966899i \(-0.582128\pi\)
−0.255160 + 0.966899i \(0.582128\pi\)
\(984\) −6.00000 −0.191273
\(985\) 20.0000 0.637253
\(986\) −4.00000 −0.127386
\(987\) 0 0
\(988\) 2.00000 0.0636285
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) −36.0000 −1.14358 −0.571789 0.820401i \(-0.693750\pi\)
−0.571789 + 0.820401i \(0.693750\pi\)
\(992\) 0 0
\(993\) 8.00000 0.253872
\(994\) 0 0
\(995\) 32.0000 1.01447
\(996\) 4.00000 0.126745
\(997\) −54.0000 −1.71020 −0.855099 0.518465i \(-0.826503\pi\)
−0.855099 + 0.518465i \(0.826503\pi\)
\(998\) −4.00000 −0.126618
\(999\) 2.00000 0.0632772
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 5586.2.a.t.1.1 1
7.6 odd 2 798.2.a.i.1.1 1
21.20 even 2 2394.2.a.b.1.1 1
28.27 even 2 6384.2.a.m.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
798.2.a.i.1.1 1 7.6 odd 2
2394.2.a.b.1.1 1 21.20 even 2
5586.2.a.t.1.1 1 1.1 even 1 trivial
6384.2.a.m.1.1 1 28.27 even 2